Determine the number of roots, counting multiplicities, of the equation

in the annulus 1 ≤ |z| < 2.

Step-by-Step Solution:

Chapter:Problem:

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Step 1 of 5

Rouche’s Theorem:

Suppose
and
are two analytic function inside and on to a simple closed contour
and satisfies
for each point on
.Then f (z) and f (z) +g (z) have the same number of zeroes,
counting multiplicities inside C .

To find the number of roots, counting multiplicities of
in the annulus
, first calculate the number of zeros inside the circle
of

Suppose:

Consider,

And
has zero with multiplicity 5.

Now consider,

Clearly,

So, by Rouche’s theorem,

has zero with counting multiplicity 5 on and inside the
circle
=
…… (1)